What would be the exact functionality of a capacitor? Answered

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Think of it like a rubber band. Apply voltage and it stretches, storing that energy. When the voltage is disconnected or drops, some (or all) of the energy is released.
A good example is in power supplies. When there's a sudden current draw, the supply voltage might drop temporarily. The supply voltage drops, which releases some of the energy stored in the capacitor and picks up the slack.
They're also used in circuits to block DC current, imagine stretching a rubber band to its limits, and it just won't stretch any more. From that point on, only changes in voltage will make it through the other side of the capacitor.

think of capacitor(as a water basin) and voltage (as water) ..........water(voltage) flows into the basin (capacitor) until basin is full.. then the basin releases water at a selected rate to keep the river (voltage )running smoothly ......like an a/c startup capacitor if a/c did not have this to hold a large voltage charge for a/c compressor to use to start ..... the voltage demand from the compressor without a capacitor would trip the homes circut breaker or blow a fuse each time it tried to start-up

Start with basic electronics : )
AC is alternating current-pulses or waves of power that go from, say, 120 volts to zero volts 60 times a second. DC is constant voltage, not changing.
If you want to make AC (house electricity) into DC (battery electricity) you have to even out the bumps-that's one use of a capacitor. It stores some of the electricity so that it can 'fill in' the AC electricity.

I like to think of them as barrels of water. The higher the capacitance the bigger the barrel, the higher the voltage the higher the barrel is raised (increase of potential). But that is only in the sense of using them as a storage device they can also be used to pass or shunt signals as in a filter.
It really depends on the circuit it is being used in.

Well here's how it works; (Or at least how I had it explained to me)
The electrons flow into the capacitor, and stay there, like a big parking lot.
Then, when there's enough, they all flow out the other end to provide electricity to the other side of the circuit. It's sort of like a small battery.

Basically a capacitor can be used to act as a reservoir to store charge or can be used to block DC and pass AC. It will also change the voltage / current phase of an AC signal.
Im not sure what you mean by "saving a little energy", please explain and I will try to tell you why